This presentation has a character of a review of recent results in the field of constructing Clar covers of graphene flakes and using them for determination of various molecular properties of these molecules. Starting with a definition of a Clar cover (i.e., an extended Kekulé structure), we show how to construct the totality of Clar covers for an analyzed graphene flake and how to obtain their total number without their explicit construction. The resulting theory has a beautiful mathematical structure with connections to various fields of mathematics: graph theory, continued fraction theory, order theory, and combinatorics with particular focus on the enumeration of linear extensions of a certain family of partially ordered sets. Chemical applications of the presented theory will focus on the distribution of aromaticity patterns in graphene flakes and on the determination of spin populations in singly, doubly, and triply radical graphene flakes.